Dynamical Analysis Of A Predator-prey System With Generalized Holling Type ? Functional Response

A predator-prey model with generalized Holling type IV functional response?(x)=mx/ax~2+bx+1and Leslie type numerical response is studied in this paper.In order to describe the relations more precisely between the predator and prey in the reality,the range of the parameter b is extended to(-2(?),+?),while the traditional range of the parameter b is[0,+?).In this case,the generalized Holling type IV functional response?(x)becomes a nonmonotonic function.Correspondingly,there are very complicated dynamical phenomena in the predator-prey model with generalized Holling type IV functional response.The dynamical properties of the predator-prey model is researched in this paper.For some values of parameters,there are two nonhyperbolic positive equilibrium(one is a multiple focus of multiplicity one,the other is a cusp of codimension 2).For some other values of parameters,there is only one degenerate equilibrium which is a Bogdanov-Takens singularity of codimension 3.The bifurcation analysis is made in this paper for these two cases.In the case of two nonhyperbolic positive equilibrium,the model undergoes a Bogdanov-Takens bifurcation and a Hopf bifurcation in the small neighborhoods of these two equilibrium.The numerical experiments show that the model can have the folling cases:(1)one stable limit cycle enclosing two equilibrium;(2)one unstable limit cycle enclosing one hyperbolic equilibrium;(3)two limit cycle enclosing one equilibrium;(4)one stable limit cycle enclosing a homoclinic loop;(5)three equilibrium enclosed by a stable limit cycle;(6)two stable equilibrium and one stable limit cycle.In the case of unique degenerate Bogdanov-Takens singularity,the existence of the degenerate focus type Bogdanov-Takens bifurcation of codimension 3 is proved.